in modified Bessel functions

SearchAdvancedHelp

(0.034 seconds)

21—30 of 111 matching pages

21: 10.34 Analytic Continuation

§10.34 Analytic Continuation

►When $m\in \mathbb{Z}$, … ►

10.34.2 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{-m\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)-\pi \mathrm{i}\sin \left(m\nu \pi \right)\csc \left(\nu \pi \right)I_{\nu }\left(z\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.31

Referenced by:

§10.27, §10.34, §10.34

Permalink:

http://dlmf.nist.gov/10.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

… ►If $\nu =n(\in \mathbb{Z})$, then limiting values are taken in (10.34.2) and (10.34.4): ►

10.34.5 $$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}{K}_n\left(z\right)+{(-1)}^{n(m-1)-1}m\pi \mathrm{i}{I}_n\left(z\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

…

22: 10.28 Wronskians and Cross-Products

§10.28 Wronskians and Cross-Products

►

10.28.1 $$𝒲\left\{I_{\nu }\left(z\right),I_{-\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)I_{-\nu -1}\left(z\right)-I_{\nu +1}\left(z\right)I_{-\nu }\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

ⓘ

Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.14

Permalink:

http://dlmf.nist.gov/10.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

►

10.28.2 $$𝒲\left\{K_{\nu }\left(z\right),I_{\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)K_{\nu +1}\left(z\right)+I_{\nu +1}\left(z\right)K_{\nu }\left(z\right)=1/z.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.15

Permalink:

http://dlmf.nist.gov/10.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

23: 10.41 Asymptotic Expansions for Large Order

… ► …

24: 2.8 Differential Equations with a Parameter

… ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … ►For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). …

25: 10.29 Recurrence Relations and Derivatives

… ►

§10.29(i) Recurrence Relations

►With ${𝒵}_{\nu }\left(z\right)$ defined as in §10.25(ii), … ►

$I_0^{\prime }\left(z\right)=I_1\left(z\right),$

… ►For results on modified quotients of the form $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). ►

§10.29(ii) Derivatives

…

26: 10.43 Integrals

… ►

10.43.4 $${\int }_0^x\frac{I_0\left(t\right)-1}{t}dt=\frac{1}{2}\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{\psi \left(k+1\right)-\psi \left(1\right)}{k!}{(\frac{1}{2}x)}^k{I}_k\left(x\right)=\frac{2}{x}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k(2k+3)(\psi \left(k+2\right)-\psi \left(1\right))I_{2k+3}\left(x\right).$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer and $x$: real variable

Referenced by:

§10.43(ii)

Permalink:

http://dlmf.nist.gov/10.43.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(ii), §10.43 and Ch.10

►

10.43.5 $${\int }_x^{\mathrm{\infty }}\frac{K_0\left(t\right)}{t}dt=\frac{1}{2}{\left(\ln \left(\frac{1}{2}x\right)+\gamma \right)}^2+\frac{{\pi }^2}{24}-\sum _{k=1}^{\mathrm{\infty }}\left(\psi \left(k+1\right)+\frac{1}{2k}-\ln \left(\frac{1}{2}x\right)\right)\frac{{(\frac{1}{2}x)}^{2k}}{2k{(k!)}^2},$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $x$: real variable

A&S Ref:

11.1.22

Referenced by:

§10.43(ii)

Permalink:

http://dlmf.nist.gov/10.43.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(ii), §10.43 and Ch.10

…

27: 10.56 Generating Functions

… ►

10.56.3 $$\frac{\cosh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\cosh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(1)}\left(z\right),$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

►

10.56.4 $$\frac{\sinh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\sinh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(2)}\left(z\right),$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

►

10.56.5 $$\frac{\exp \left(-\sqrt{z^2+2\mathrm{i}zt}\right)}{z}=\frac{{\mathrm{e}}^{-z}}{z}+\frac{2}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{{(-\mathrm{i}t)}^n}{n!}{𝗄}_{n-1}\left(z\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56, Ch.10

Permalink:

http://dlmf.nist.gov/10.56.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

28: 18.34 Bessel Polynomials

§18.34 Bessel Polynomials

►

§18.34(i) Definitions and Recurrence Relation

… ►where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … ►expressed in terms of Romanovski–Bessel polynomials, Laguerre polynomials or Whittaker functions, we have … ►For uniform asymptotic expansions of $y_n(x;a)$ as $n\to \mathrm{\infty }$ in terms of Airy functions (§9.2) see Wong and Zhang (1997) and Dunster (2001c). …

29: 11.4 Basic Properties

… ►

11.4.2 $${𝐋}_{n+\frac{1}{2}}\left(z\right)=I_{-n-\frac{1}{2}}\left(z\right)-{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sum _{m=0}^n\frac{{(-1)}^m(2m)!{\mathrm{\hspace{0.17em}2}}^{-2m}}{m!(n-m)!}{(\frac{1}{2}z)}^{n-2m},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $n$: integer order

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

… ►

§11.4(iv) Expansions in Series of Bessel Functions

…

30: 10.66 Expansions in Series of Bessel Functions

… ►

10.66.1 $${\mathrm{ber}}_{\nu }x+\mathrm{i}{\mathrm{bei}}_{\nu }x=\sum _{k=0}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{(3\nu +k)\pi \mathrm{i}/4}{x}^k{J}_{\nu +k}\left(x\right)}{2^{k/2}k!}=\sum _{k=0}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{(3\nu +3k)\pi \mathrm{i}/4}{x}^k{I}_{\nu +k}\left(x\right)}{2^{k/2}k!}.$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.32

Referenced by:

§10.66

Permalink:

http://dlmf.nist.gov/10.66.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.66 and Ch.10

…